Are scattering amplitudes dual to super Wilson loops?

The MHV scattering amplitudes in planar N=4 SYM are dual to bosonic light-like Wilson loops. We explore various proposals for extending this duality to generic non-MHV amplitudes. The corresponding dual object should have the same symmetries as the scattering amplitudes and be invariant to all loops under the chiral half of the N=4 superconformal symmetry. We analyze the recently introduced supersymmetric extensions of the light-like Wilson loop (formulated in Minkowski space-time) and demonstrate that they have the required symmetry properties at the classical level only, up to terms proportional to field equations of motion. At the quantum level, due to the specific light-cone singularities of the Wilson loop, the equations of motion produce a nontrivial finite contribution which breaks some of the classical symmetries. As a result, the quantum corrections violate the chiral supersymmetry already at one loop, thus invalidating the conjectured duality between Wilson loops and non-MHV scattering amplitudes. We compute the corresponding anomaly to one loop and solve the supersymmetric Ward identity to find the complete expression for the rectangular Wilson loop at leading order in the coupling constant. We also demonstrate that this result is consistent with conformal Ward identities by independently evaluating corresponding one-loop conformal anomaly.     

Chiral symmetry breakdown,anomaly, and the PCAC relation on a lattice

Lattice fermion formulation is investigated using a solvable model which resembles quantum chromodynamics. CP₂^N?1 models with quarks are formulated on a lattice. For dynamical quarks, a generalized formulation of the Wilson and the Osterwalder-Seiler lattice fermion is used. In the $\text{1}\text{N}$ expansion, the spontaneous breakdown of chiral symmetry (which is softly broken by the quark mass) apparently occurs in this model, and the “pion” mass is calculated. From the above results, it is shown that the above lattice fermion formulations have the desired continuum limit. The axial-vector current is investigated and it is proved that the usual anomaly appears in the continuum limit and the PCAC relation is satisfied.     

Index of a family of lattice Dirac operators and its relation to the non-Abelian anomaly on the lattice

In the continuum, a topological obstruction to the vanishing of the non-Abelian anomaly in 2n dimensions is given by the index of a certain Dirac operator in 2n+2 dimensions, or equivalently, the index of a 2-parameter family of Dirac operators in 2n dimensions. In this paper an analogous result is derived for chiral fermions on the lattice in the overlap formulation. This involves deriving an index theorem for a family of lattice Dirac operators satisfying the Ginsparg-Wilson relation. The index density is proportional to Lüscher's topological field in 2n+2 dimensions.     

Chiral symmetry on the lattice with Wilson fermions

The chiral properties of the continuum limit of lattice QCD with Wilson fermions are studied. We show that a partially conserved axial current can be defined, satisfying the usual current algebra requirements.A proper definition of the chiral symmetry order parameter, $\text{〈0|}\text{ψ}\text{ψ|0〉}$, is given, and the chiral properties of composite operators are investigated. The implications of our analysis to the lattice determination of non-leptonic weak amplitudes are also discussed.     

Space-time approach to anomalies in the W-T identities and commutators

The Ward-Takahashi identities of the axial-type n-point functions are analysed, assuming the conformal invariance and the chiral SU(3) × SU(3) symmetry to become exact at the short distance. The anomalies in the identities are obtained in a simple nonperturbative way, and the commutator anomalies are determined in connection with the pinching mechanism in taking the equal-time limit of commutators.     

Wilson fermions in a two-dimensional model

Wilson's action for fermions on a lattice is compared to the continuum action in a model obtained from the chiral Gross-Neveu model by performing a chiral transformation. The local definition of the axial current leads to two anomalies unrelated by the constraint of Lorentz invariance. In the large-N limit, the mass counterterm of the action is determined; this term is unnecessary in the Osterwalder-Seiler regularization. An expansion in the fermion propagator and in the axial current coupling may be formulated and summed to all orders for large N.     

Non-perturbative renormalization of lattice four-fermion operators without power subtractions

A general nonperturbative analysis of the renormalization properties of four-fermion operators in the framework of lattice regularization with Wilson fermions is presented. We discuss the nonperturbative determination of the operator renormalization constants in the lattice regularization independent (RI or MOM) scheme. We also discuss the determination of the finite lattice subtraction coefficients from Ward identities. We prove that, at large external virtualities, the determination of the lattice mixing coefficients, obtained using the RI renormalization scheme, is equivalent to that based on Ward identities, in the continuum and chiral limits. As a feasibility study of our method, we compute the mixing matrix at several renormalization scales, for three values of the lattice coupling , using the Wilson and tree-level improved SW-Clover actions. Received: 26 February 1999 / Published online: 15 July 1999     

Parafermionic theory with the symmetry Z5

A parafermionic conformal theory with the symmetry Z₅ is constructed, based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra.The primary operators of the theory, which are the singlet, doublet 1, doublet 2, and disorder operators, are found to be accommodated by the weight lattice of the classical Lie algebra B₂. The finite Kac tables for unitary theories are defined and the formula for the conformal dimensions of primary operators is given.     

The supersymmetric Ward identities on the lattice

Supersymmetric (SUSY) Ward identities are considered for the N=1 SU(2) SUSY Yang-Mills theory discretized on the lattice with Wilson fermions (gluinos). They are used in order to compute non-perturbatively a subtracted gluino mass and the mixing coefficient of the SUSY current. The computations were performed at gauge coupling and hopping parameter , 0.194, 0.1955 using the two-step multi-bosonic dynamical-fermion algorithm. Our results are consistent with a scenario where the Ward identities are satisfied up to O(a) effects. The vanishing of the gluino mass occurs at a value of the hopping parameter which is not fully consistent with the estimate based on the chiral phase transition. This suggests that, although SUSY restoration appears to occur close to the continuum limit of the lattice theory, the results are still affected by significant systematic effects. Received: 8 November 2001 / Revised version: 14 January 2001 / Published online: 15 March 2002     

Reduced Dirac-Kähler lattice fermion action

It is shown that the lattice Dirac-Kähler action is reducible under a chiral-like transformation. This provides a new lattice fermion action for spinors that have 2^d−1 components (instead of 2^d), with the property that, in the free case, each component satisfies the lattice euclidean Klein-Gordon equation. Reflection positivity is satisfied on the lattice, thus assuring a (positive) physical Hilbert space. In d = 4 dimensions the spinors have 8 components, and the correct physical chiral anomaly in the continuum limit. The action is suitable for QCD quarks which, in the continuum limit, are described by Dirac spinors that occur in flavor doublets.     

Overlap valence quarks on a twisted mass sea: A case study for mixed action lattice QCD

We discuss a lattice QCD mixed action investigation employing Wilson maximally twisted mass sea and overlap valence fermions. Using four values of the lattice spacing, we demonstrate that the overlap Dirac operator assumes a point-like locality in the continuum limit. We also show that by adopting suitable matching conditions for the sea and valence theories a consistent continuum limit for the pion decay constant and light baryon masses can be obtained. Finally, we confront results for sea–valence mixed meson masses and the valence scalar correlator with corresponding expressions of chiral perturbation theory. This allows us to extract low energy constants of mixed action chiral perturbation which characterize the strength of unitarity violations in our mixed action setup.     

The Wess-Zumino term in lattice theories

The chiral property of the Wilson lattice fermion is investigated. A chiral-invariant four-Fermi model, in which chiral symmetry is dynamically broken, is considered in 2 and 4 dimensions. The Wess-Zumino term is calculated in the 1/N_c expansion. In 2 dimensions, the Wess-Zumino term appears from the Wilson term in the desired form. However, in 4 dimensions the mass-independent one does not appear. The physical reason for this result is discussed.     

Heterotic Toda fields

A non-left-right symmetric conformal integrable Toda field theory is constructed. It is found that the conformal algebra for this model is the product of a left chiral W_r+1 algebra and a right chiral W_r+1² algebra. The general classical solution is constructed out of the chiral vectors satisfying the so-called classical exchange algebra. In addition, we derived an explicit Wronskian type solution in relation to the constrained WZNW theory. We also showed that the A_∞ limit of this model is precisely the (B₂, C₁) flow of the standard Toda lattice hierarchy.     

Observations on the Wilson fermions in the ϵ regime

We make several observations concerning the low quark mass region with Wilson fermions and how this is connected with the ? regime in the continuum. A transition from tiny cutoff effects to rather large discretization errors would take place in general with Wilson fermions if we lower the quark mass at finite lattice spacing. We argue that these two regions exhibit rather different behaviours concerning the coupling between cutoff effects and zero-modes. We interpolate between these two regimes adding to the continuum ? regime formulae, in the spirit of the Symanzik expansion, the relevant operators parametrising the leading cutoff effects. We compute the partition function, the chiral condensate, scalar and pseudo-scalar correlation functions. The final formulae can be used to fit lattice data to extract physical low energy constants, and to estimate systematic uncertainties coming from discretization errors. Moreover they suggest ways on how to remove these cutoff effects, the core of which are captured by the continuum zero modes.     

Kac-Moody symmetries of critical ground states

The symmetries of critical ground states of two-dimensional lattice models are investigated. We show how mapping a critical ground state to a model of a rough interface can be used to identify the chiral symmetry algebra of the conformal field theory that describes its scaling limit. This is demonstrated in the case of the six-vertex model, the three-coloring model on the honeycomb lattice, and the four-coloring model on the square lattice. These models are critical and they are described in the continuum by conformal field theories whose symmetry algebras are the su(2)_k=1, su(3)_k=1, and the su(4)_k=1 Kac-Moody algebra, respectively. Our approach is based on the Frenkel-Kac-Segal vertex operator construction of level-one Kac-Moody algebras.     

Fermion theories on a 2d torus with Wilson action improved by Pauli – Villars regularization

Vectorial and anomaly free chiral U(1) fermion models on a 2d finite lattice are considered. It is demonstrated both numerically and analytically that introduction of Pauli – Villars type regularization suppresses the symmetry breaking effects caused by the Wilson term.     

Wess-Zumino Effective Action on a Lattice

Taking Wilson fermion formulation for the chiral gauge theory we derive the Wess-Zumino effective action in a lattice version in both two and four dimensions. Also the relationship between the anomalous generation and the chiral symmetry breaking is presented. Our results in continuum limit are coincident with those obtained by other regularizations.     

Conformal anomaly in 4D gravity-matter theories non-minimally coupled with a dilaton

The conformal anomaly for 4D gravity-matter theories, which are non-minimally coupled with the dilaton, is systematically studied. Special care is taken of rescaling of fields, treatment of total derivatives, hermiticity of the system operator and the choice of measure. Scalar, spinor and vector fields are taken as the matter quantum fields and their explicit conformal anomalies in the gravity-dilaton background are found. The cohomology analysis is carried out and some new conformal invariants and trivial terms, involving the dilaton, are obtained. The symmetry of the constant shift of the dilaton field plays an important role. The general structure of the conformal anomaly is examined. It is shown that the dilaton affects the conformal anomaly characteristically for each case: (1) [Scalar] The dilaton changes the conformal anomaly only by a new conformal invariant, I₄; (2) [Spinor] The dilaton does not change the conformal anomaly; (3) [Vector] The dilaton changes the conformal anomaly by three new (generalized) conformal invariants, I₄, I₂, I₁. We present some new anomaly formulae which are useful for practical calculations. Finally, the anomaly induced action is calculated for the dilatonic Wess-Zumino model. We point out that the coefficient of the total derivative term in the conformal anomaly for the 2D scalar coupled to a dilaton is ambiguous. This resolves the disagreement between earlier calculations and the result of Hawking and Bousso.